Is the word demonstrar the same in Brazilian and European Portuguese?

Yes

The word "demonstrar" is identical in spelling, meaning, and grammar in both Brazilian and Continental Portuguese. The differences are strictly phonetic. In Brazilian Portuguese, vowels are generally more open and the rhythm of speech is more melodic and "syllable-timed." In Continental Portuguese, unstressed vowels are often reduced or even suppressed (elided), and the rhythm is more "staccato" or consonant-heavy. Additionally, the pronunciation of the "r" at the end of the word varies significantly; in many parts of Brazil, it may be aspirated or retroflex, whereas in Portugal, it is typically a more distinct uvular or tapped sound.

Brazilian Portuguese Examples

1. Ele demonstrou muita coragem durante o susto. (He showed a lot of courage during the fright.)
2. Você não demonstrou nenhum interesse pelo projeto. (You didn't show any interest in the project.)
3. O professor demonstrou como resolver o problema. (The professor demonstrated how to solve the problem.)
4. Ela demonstrou estar muito feliz com a notícia. (She showed she was very happy with the news.)
5. Eles demonstraram total apoio à decisão. (They demonstrated total support for the decision.)

Continental Portuguese Examples

1. Ele demonstrou muita coragem durante o susto. (He showed a lot of courage during the fright.)
2. Tu não demonstraste qualquer interesse pelo projeto. (You didn't show any interest in the project.)
3. O professor demonstrou como resolver o problema. (The professor demonstrated how to solve the problem.)
4. Ela demonstrou estar muito contente com a notícia. (She showed she was very happy with the news.)
5. Eles demonstraram total apoio à decisão. (They demonstrated total support for the decision.)